Separating double rays in locally finite planar graphs
The authors consider locally finite, 1-ended graphs \(\Gamma\) such that the automorphism group of \(\Gamma\) acts with finitely many orbits on \(V(\Gamma)\). They first show that if \(\Gamma\) is 3-connected and planar, then for each double ray \(D\) in \(\Gamma\) the graph \(\Gamma- V(D)\) has at most two infinite components and exactly two, if in addition \(D\) is a quasi-axis (i.e. \(\text{dist}_D(x, y)\leq \text{Sdist}_\Gamma(x, y)\) for some constant S and all \(x, y\in V(D)\)). Adapting a method of \textit{C. Thomassen} [The Hadwiger number of infinite vertex-transitive graphs, Combinatorica 12, No. 4, 481-491 (1992; Zbl 0776.05053)] they prove that conversely almost 4-connected \(\Gamma\) are planar if for each quasi-axis \(D\) the graph \(\Gamma- V(D)\) has exactly two infinite components.
scientific article; zbMATH DE number 810255
scientific article